Solve for x log of 2+x- log of x-2 = log of 3

$\log (2 + x) - \log (x - 2) = \log (3)$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{2 + x}{x - 2}) = \log (3)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{2 + x}{x - 2} = 3$

Step 3

Solve for $x$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 2, 1$

Remove parentheses.

$x - 2, 1$

The LCM of one and any expression is the expression.

$x - 2$

Multiply each term in $\frac{2 + x}{x - 2} = 3$ by $x - 2$ to eliminate the fractions.

Multiply each term in $\frac{2 + x}{x - 2} = 3$ by $x - 2$ .

$\frac{2 + x}{x - 2} (x - 2) = 3 (x - 2)$

Simplify the left side.

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{2 + x}{x - 2} (x - 2) = 3 (x - 2)$

Rewrite the expression.

$2 + x = 3 (x - 2)$

Simplify the right side.

Apply the distributive property.

$2 + x = 3 x + 3 \cdot - 2$

Multiply $3$ by $- 2$ .

$2 + x = 3 x - 6$

Solve the equation.

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$2 + x - 3 x = - 6$

Subtract $3 x$ from $x$ .

$2 - 2 x = - 6$

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$- 2 x = - 6 - 2$

Subtract $2$ from $- 6$ .

$- 2 x = - 8$

Divide each term in $- 2 x = - 8$ by $- 2$ and simplify.

Divide each term in $- 2 x = - 8$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 8}{- 2}$

Divide $x$ by $1$ .

$x = \frac{- 8}{- 2}$

Simplify the right side.

Divide $- 8$ by $- 2$ .

$x = 4$

Solve for x log of 2+x- log of x-2 = log of 3

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